Animations

The animation of a circular drum

Preliminary information of the Bessel function

Plot[BesselJ[1, x], {x, 0, 18}];

we pick the second  zero of the Bessel function

FindRoot[ BesselJ[1, x]==0, {x, 7}]

In the command FindRoot the bracket {x, 7} tells MATHEMATICA to look for the zero closest to 7.

We define the following function, which is the solution of the wave equation in two dimensional with cylindrical symmetry

u[m_, nthzero__, r_, theta_, t_,omega_] :=
    BesselJ[m, nthzero r] Cos[m theta] Cos [omega t]

We need another MATHEMATICA package

Needs["Graphics`ParametricPlot3D`"]

The animation is obtained by using a Do loop Do[ f[t], {t, tinit, tmax, increment t}] and we pick 15 frames. Therefore we animate the (1, 2) normal mode of a circular drum

Do[
CylindricalPlot3D[
         u[1, 7.01559, r, theta, t,10], {r, 0, 1}, 
                {theta, 0, 2 Pi},
        PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}},
        PlotPoints->50,
        Axes -> False, Boxed ->False],
               {t, 0, 2 Pi/10, Pi/ (7 10)}]
                       

Once you have all the plots, click on the second bracket from left to right.  Now double click, and
all the plots should contract to one.  Make sure that the  bracket is shaded and move the arrow to the top of the windward click on Cell and select "  Animate Selected Graphics. "  To stop the animation just click anywhere on the notebook..

Problem 5. - Try another m and find another normal mode (2, 4).